ASYMPTOTIC BEHAVIOUR IN THE SET OF NONHOMOGENEOUS CHAINS OF STOCHASTIC

doi:10.7151/dmps.1141

ASYMPTOTIC BEHAVIOUR IN THE SET OF NONHOMOGENEOUS CHAINS OF STOCHASTIC

OPERATORS

¹

Ma lgorzata Pu lka Department of Mathematics Gda´ nsk University of Technology

ul. Gabriela Narutowicza 11/12, 80-233, Gda´ nsk, Poland e-mail: mpulka@mif.pg.gda.pl

Abstract

We study different types of asymptotic behaviour in the set of (infinite di- mensional) nonhomogeneous chains of stochastic operators acting on L

¹

(µ) spaces. In order to examine its structure we consider different norm and strong operator topologies. To describe the nature of the set of nonhomoge- neous chains of Markov operators with a particular limit behaviour we use the category theorem of Baire. We show that the geometric structure of the set of those stochastic operators which have asymptotically stationary density differs depending on the considered topologies.

Keywords: Markov operator, asymptotic stability, residuality, dense G

δ

. 2010 Mathematics Subject Classification: Primary: 47A35, 47B65;

Secondary: 60J10, 54H20.

1. Introduction

The study of chains of Markov operators has become a subject of interest in regard to their applications in many different areas of science and technology. Markov operators are commonly used to describe phenomena involving a law of conserva- tion of a certain quantity, e.g. mass, energy, the number of particles in physical or chemical processes. Typical questions appear in the context of probability

1

This paper is a part of the author’s Ph.D. thesis written under the supervision of Professor

W. Bartoszek. The author wishes to express her appreciation to Professor Bartoszek for his

advice and helpful suggestions.

theory and concern the evolution of a density (probability distribution) of such a quantity. The case when the chain is homogeneous in time is well-understood and has a comprehensive literature (cf. [7, 8, 10]). In particular, the asymptotic behaviour of iterates of Markov operators has been intensively studied. The er- godic structure of homogeneous chains is fully described including probabilistic, lattice and spectral conditions for convergence of iterates with respect to all stan- dard topologies. In the case of the class of chains nonhomogeneous in time the situation is not so transparent, since no proper notion of a stationary density can be defined (in general). Thus, in order to describe the properties of a nonhomo- geneous chain one may study its asymptotic behaviour, which is understood as the study of a ”generalized concept” of stationarity. Namely, we may ask whether there exists a common limiting density or, at least, if the influence of the state of the process at the time m on its future states decreases to zero with the passage of time. Various gradations of this asymptotic properties may be considered de- pending on the mode of convergence of the iterates of the Markov operator. In this paper we focus solely on the uniform and strong modes of convergence.

Differences between the classes of homogeneous and nonhomogeneous chains attracted the attention of probabilists in the second half of the 20th century.

For example, in [6] Iosifescu observed that unlike the homogenous case, uniform asymptotic stability (strong ergodicity) is not a ”natural” concept for nonhomo- geneous chains. Thus, given a class of possible evolutions of Markov operators, i.e., a class of nonhomogeneous chains of Markov operators with a particular asymptotic behaviour, one may ask about its topological size. Such a description is based on the category theorem of Baire. Namely, the set is recognized as a large object if it is residual (it contains a dense G

_δ

set). Thus, generic evolutions are those which belong to a residual subset. The aim of this paper is to define different types of asymptotic behaviour of nonhomogeneous chains of Markov op- erators acting on L

¹

(µ) spaces and to determine which one of them is prevalent.

The geometric structure of infinite dimensional nonhomogeneous Markov chains defined on the `

¹

space of all absolutely summable real sequences was intensively studied in [12]. Since `

¹

= L

¹

(N, 2

^N

, counting measure ), then the results included in this article are generalizations of the results obtained in [12].

For the convenience of the reader most of the theorems are proved in full detail.

This paper may be considered as the first step to generalizations of some results included in [3]. It is worth noticing that in [11] the asymptotic properties of nonhomogeneous discrete Markov processes with general state space L

¹

(µ) were studied and the results obtained were applied in the investigation of the limit be- haviour of the so-called quadratic stochastic processes which are concerned with genetic models.

Let (X, A, µ) be a separable σ-finite measure space. Throughout the paper

we consider the (separable) Banach lattice of real and A-measurable functions

f such that |f | is µ-integrable and we denote it by L

¹

(µ). By k · k

1

we denote the relevant norm. We say that a linear operator P : L

¹

(µ) → L

¹

(µ) is Markov (or stochastic) if

P f ≥ 0 and kP f k

₁

= kf k

₁

for all f ≥ 0, f ∈ L

¹

(µ). By D = D(X, A, µ) we denote the set of all densities on X, i.e.,

D = f ∈ L

¹

(µ) : f ≥ 0, kf k

₁

= 1 .

In view of stochasticity of P we have that k|P |k = 1 (where k| · |k stands for the norm operator) and P (D) ⊂ D. The sequence of such operators denoted by P := (P

^m,m+1

)

m≥0

is called a discrete time nonhomogeneous chain of Markov operators. For any natural numbers 0 ≤ m < n we set

P

^m,n

= P

^m,m+1

◦ P

^m+1,m+2

◦ · · · ◦ P

^n−1,n

.

If for each m ≥ 0 one has P

^m,m+1

= P , then P = (P )

_m≥0

is called a homogeneous chain of Markov operators. The set of all chains of Markov operators (including homogeneous) will be denoted by S , i.e.,

S = n

P = P

^m,m+1

m≥0

: P

^m,m+1

are Markov operators o .

Let t ∈ [0, 1] be given. A convex combination T(t) of two chains of Markov operators P and R ∈ S is defined as follows:

T

^m,m+1

(t) = tP

^m,m+1

+ (1 − t) R

^m,m+1

.

Note that T(t) ∈ S for every t ∈ [0, 1] and that a mapping [0, 1] 3 t 7→ T(t) ∈ S is continuous when S is equipped with suitable topology. Moreover, T(0) = R and T(1) = P. Thereby, S has an affine structure and it is arcwise connected.

Throughout the paper we write N

0

= N ∪ {0}.

We will endow the set S with metric structures. Given P, R ∈ S let us consider:

(1) the sup norm operator topology induced by the metric d

_{n. sup}

(P, R) = sup

m

 P

^m,m+1

− R

^m,m+1

  , (2) the P norm operator topology induced by the metric

d

_n.^P

(P, R) =

∞

X

m=0

1 2

^m+1

 P

^m,m+1

− R

^m,m+1





,

(3) the P sup strong operator topology induced by the metric d

_{so. sup}

(P, R) =

∞

X

l=0

1 2

^l

sup

m

P

^m,m+1

f

_l

− R

^m,m+1

f

_l

1

,

where {f

0

, f

1

, . . .} is a fixed countable and linearly dense subset of D, (4) the P P strong operator topology induced by the metric

d

_so.^P

(P, R) =

∞

X

m,l=0

1 2

^m+l+1

P

^m,m+1

f

l

− R

^m,m+1

f

l

1

,

where {f

0

, f

1

, . . .} is a fixed countable and linearly dense subset of D.

Note that d

_{so. sup}

(P

_k

, R) → 0 as k → ∞ if and only if for every f ∈ L

¹

(µ) (f ∈ D) and any m ∈ N

0

one has lim

_k→∞

sup

_m

kP

_k^m,m+1

f − R

^m,m+1

f k

₁

= 0.

Moreover, the topologies generated by d

so. sup

and d

_so.^P

do not depend on the choice of a sequence {f

₀

, f

₁

, . . .}.

Clearly, d

_{n. sup}

generates the strongest topology and d

_so.^P

generates the weakest. However, it should be emphasized that metrics d

_n.^P

and d

so. sup

are not comparable. In order to observe it, consider P

_j

= (P

_j^m,m+1

)

_m≥0

∈ S defined as follows:

P

_j^m,m+1

=

( P, if 0 ≤ m < j, I, if m ≥ j,

where I stands for the identity operator and P = (P )

m≥0

is such that P 6= I.

Then

d

_n.^P

(P

j

, P) =

∞

X

m=0

1 2

^m+1

 

 P

_j^m,m+1

− P

^m,m+1

  

=

j−1

X

m=0

1

2

^m+1

k|P − P |k +

∞

X

m=j

1

2

^m+1

k|I − P |k

= 1

2

^j

k|I − P |k → 0 as j → ∞.

On the other hand, for a given fixed countable and dense subset {f

0

, f

1

, . . .} of D we have

d

so. sup

(P

j

, P) =

∞

X

l=0

1 2

^l

sup

m

P

_j^m,m+1

f

l

− P

^m,m+1

f

l

1

=

∞

X

l=0

1

2

^l

kf

_l

− P f

_l

k

₁

> 0.

Thus d

so. sup

(P

j

, P) 9 0 as j → ∞. Thereby, d

n.P

is not stronger than d

so. sup

. Now we shall see that d

so. sup

is not stronger than d

_n.^P

. In order to prove it observe that since the measure µ is σ-finite, there exists a sequence {B

_k

}, B

_k

∈ A, such that B

i

∩ B

_j

= ∅ for i 6= j and

X =

∞

[

k=0

B

_k

, 0 < µ (B

_k

) < ∞ for all k ∈ N

0

.

Let g

k

∈ D be such that the essential support supp g

_k

:= {x ∈ X : g

k

(x) 6= 0} ⊆ B

_k

for any k ∈ N

0

. For any f ∈ L

¹

(µ) denote

a

_k

(f ) = Z

Bk

f dµ.

Note that P

∞

k=0

a

_k

(f ) = 1 if f ∈ D. Then we can define P

_j

= (P

_j^m,m+1

)

_j≥0

∈ S as follows:

P

_j^m,m+1

f = P

_j

f = f 1

^Sj

k=0Bk

+

∞

X

k=j+1

a

_k

(f ) · g

₀

.

Let I = (I, I, . . .) ∈ S , where as before I stands for the identity operator. Then

d

so. sup

(P

j

, I) =

∞

X

l=0

1 2

^l

sup

m

P

_j^m,m+1

f

l

− If

_l

1

=

∞

X

l=0

1

2

^l

kP

_j

f

_l

− f

_l

k

₁

=

∞

X

l=0

1 2

^l

f

l

1

^Sj

k=0Bk

+

∞

X

k=j+1

a

k

(f

l

) g

0

− f

_l

1

^Sj

k=0Bk

− f

_l

1

^S∞

k=j+1Bk

1

≤

∞

X

l=0

1 2

^l





∞

X

k=j+1

a

_k

(f

_l

) kg

₀

k

₁

+ f

_l

1

^S∞

k=j+1Bk

1





=

∞

X

l=0

1 2

^l





∞

X

k=j+1

a

_k

(f

_l

) + Z

S∞ k=j+1Bk

f

_l

dµ





=

∞

X

l=0

1 2

^l

· 2

∞

X

k=j+1

a

k

(f

l

)

=

∞

X

l=0

1

2

^l−1

1 −

j

X

k=1

a

k

(f

l

)

!

→ 0 as j → ∞.

On the other hand, d

_n.^P

(P

j

, I) =

∞

X

m=0

1

2

^m+1

k|P

_j^m,m+1

− I|k

=

∞

X

m=0

1 2

^m+1

!

k|P

_j

− I|k = 1 · 2 = 2 9 0 as j → ∞.

Therefore d

so. sup

is not stronger than d

_n.^P

. It follows that the metrics d

_n.^P

and d

_{so. sup}

are not comparable. The relationships between the considered metrics are illustrated in the Figure 1.

4

4 4 4

d

n. sup

d

_{so. sup}

d

_so.^P

d

_n.^P

Figure 1. The relationships between the metrics d

_{n. sup}

, d

_n.^P

, d

_{so. sup}

, d

_so.^P

.

Let us note that in the class of homogeneous chains of Markov operators, metrics d

n. sup

and d

_n.^P

are equivalent. In fact, if P = (P )

m≥0

, R = (R)

m≥0

∈ S , then d

n. sup

(P, R) = d

_n.^P

(P, R) = k|P − R|k. Similarly we find that in the homogeneous case metrics d

_{so. sup}

and d

_so.^P

are equivalent and for any P = (P )

_m≥0

, R = (R)

_m≥0

∈ S one has d

so. sup

(P, R) = d

_so.^P

(P, R) = P

∞

l=0 1

2^l

kP f

_l

− Rf

_l

k

₁

, where {f

₀

, f

₁

, . . .} is a fixed countable and linearly dense subset of D. This supports our remark that the nonhomogeneous case is more complex than the homogeneous one.

In what follows we study different types of asymptotic behaviour of nonho-

mogeneous chains of stochastic operators as well as residualities in the set S .

We shall see that the geometric structure of the set of those stochastic operators

which have asymptotically stationary density differs depending on the considered

topologies. We prove that the set of those Markov operators which do not possess

limiting density is dense and its interior is nonempty in the topology induced by

the metric d

_{n. sup}

. On the other hand, it occurs that the set of those operators

for which the limiting density exists is dense while S is endowed with topology

induced by the metric d

_n.^P

. We also examine the set of Markov operators which

we call (uniformly or strongly, if studied in norm or strong operator topology

respectively) almost asymptotically stable and we prove that it forms a residual

subset for both norm and strong operator topologies.

Note that d

n. sup

is the most relevant metric (topology) in studying the limit be- haviour of nonhomogeneous chains of stochastic operators. It should be empha- sized that, in contrast to the homogeneous case, the property of denseness of the set of nonhomogeneous chains of stochastic operators with a particular asymp- totic behaviour does not suffice to understand its ”size”. It derives from the fact that in the case of P sup and P P strong operator topologies the denseness of the complement of the set mentioned above can always be proved by modifying on the tail so-called sweeping operators or a fixed stochastic projection. There- fore, in order to describe the nature of the set we use the category theorem of Baire. This is because the space S equipped with any of the metrics (1)–(4) is complete and the classical Baire theorem is applicable.

The Baire category of asymptotic stability for homogeneous Markov chains was worked out in e.g. [2, 9, 13]. It should be clearly understood that our results are not a direct analogy of what was obtained in these works. In particular, it was proved in [13] that uniformly asymptotically stable (quasi-compact) ho- mogeneous Markov chains form a dense G

δ

subset in norm operator topology.

In our nonhomogeneous case, the set of those chains of operators which are not uniformly asymptotically stable has a nonempty interior.

There are more relevant works in the literature dealing with the topic of the limit behaviour of nonhomogeneous chains of Markov operators. The reader should be warned that authors do not always use the same names for the same notions. For example, in [5] and [6] strong ergodicity is what we call uniform asymptotic stability and weak ergodicity is what we refer to as almost uniform asymptotic stability. Some authors apply the terminology derived from the er- godic theory to the theory of stochastic processes and denominate what we call asymptotic stability by mixing (cf. [2, 3, 9, 12]). See [5] for detailed classification of different types of asymptotic behaviour of nonhomogeneous Markov chains.

2. Uniform asymptotic stability

In this section we examine the strongest case of asymptotic stability of chains of Markov operators, i.e., uniform asymptotic stability. We start with

Definition. A nonhomogeneous chain of Markov operators P is called uniformly asymptotically stable if there exists a unique f

∗

∈ D such that for every m ∈ N

0

n→∞

lim sup

f ∈D

kP

^m,n

f − f

∗

k

₁

= 0.

The set of all uniformly asymptotically stable chains of Markov operators is de-

noted by S

uas

.

Note that uniformly asymptotically stable chains of operators possess common limiting density and the mode of convergence is uniform. The following theorem is concerned with the prevalence problem in the set S .

Theorem 1. The set S

uas^c

of all Markov operators which are not uniformly asymptotically stable is a sup norm topology dense subset of S (i.e., in d

n. sup

).

Moreover, in this case its interior IntS

_uas^c

6= ∅.

Proof. Let P ∈ S and 0 < ε < 1 be taken arbitrarily. As before, since the measure µ is σ-finite, there exists a sequence {B

k

}, B

_k

∈ A, such that B

_i

∩B

_j

= ∅ for i 6= j and

X =

∞

[

k=0

B

_k

, 0 < µ (B

_k

) < ∞ for all k ∈ N

0

.

Let g

_k

∈ D be such that supp g

_k

= {x ∈ X : g

_k

(x) 6= 0} ⊆ B

_k

for any k ∈ N

0

. Then we can define R ∈ S as follows: for any f ∈ L

¹

(µ),

R

^m,m+1

f =

∞

X

j=m+1

a

_j−m−1

(f ) · g

_j

,

where a

j

(f ) = R

Bj

f dµ. Note that P

∞

j=0

a

j

(f ) = 1 if f ∈ D. Consider a convex combination

P

_ε^m,m+1

= (1 − ε) P

^m,m+1

+ εR

^m,m+1

. Clearly, P

ε

= (P

ε^m,m+1

)

m≥0

∈ S . We have

d

n.sup

(P

ε

, P) = sup

m

 (1 − ε) P

^m,m+1

+ εR

^m,m+1

− P

^m,m+1

 

= ε sup

m

 P

^m,m+1

− R

^m,m+1

  ≤ 2ε.

It remains to show that P

ε

is not uniformly asymptotically stable. Suppose that, on the contrary, there exists f

∗

∈ D such that for every f ∈ D we have lim

n→∞

P

ε^m,n

f = f

∗

. Since f ∈ D, there exists M ∈ N

0

such that

Z

M

S

k=0

Bk

f

∗

dµ > 1 − ε.

Hence

Z

M

S

k=0

Bk

P

_ε^m,n

f dµ −−−→

^n→∞

Z

M

S

k=0

Bk

f

∗

dµ > 1 − ε.

On the other hand, if n > m > M , then Z

M

S

k=0

Bk

P

_ε^m,n+1

f dµ = 1 − Z

∞

S

k=M +1

Bk

P

_ε^m,n+1

f dµ

= 1 − Z

∞

S

k=M +1

Bk

(1 − ε) P

^n,n+1

+ εR

^n,n+1

(P

_ε^m,n

f ) dµ

≤ 1 − ε Z

∞

S

k=M +1

Bk

R

^n,n+1

(P

_ε^m,n

f ) dµ = 1 − ε.

It follows that S

uas^c

is d

_{n. sup}

dense in S .

To show that Int S

uas^c

6= ∅ for the sup norm topology (i.e., in d

_{n. sup}

) consider the open ball

K(R, 1) := {T ∈ S : d

n. sup

(T, R) < 1} ,

where R is defined as before. If T ∈ K(R, 1), then for some 0 < ε < 1 sup

f ∈D

T

^m,m+1

f − R

^m,m+1

f

1

≤ d

_{n. sup}

(T, R) = 1 − ε.

Hence for every m + 1 > M and every f ∈ D, Z

M

S

k=0

Bk

T

^0,m+1

f dµ ≤ d

_{n. sup}

(T, R) = 1 − ε.

Thus,

sup

M ∈N

lim sup

m→∞

Z

M

S

k=0

Bk

T

^0,m+1

f dµ ≤ d

n. sup

(T, R) = 1 − ε < 1,

and therefore T has no ”invariant” densities (common limiting density). It follows that T ∈ S

uas^c

.

In the next result we shall see that topologies on S generated by d

n. sup

and d

_n.^P

differ. Namely S

uas

is large for d

_n.^P

. In fact, we have

Proposition 2. The set S

uas

is P norm topology dense in S (i.e., in d

n.P

).

Proof. Let P ∈ S and 0 < ε < 1 be taken arbitrarily. There exists M ∈ N

0

such that

₂¹M

< ε. Define P

_ε

∈ S as follows:

P

_ε^m,m+1

=

( P

^m,m+1

, if m ≤ M,

E, if m > M,

where Ef = ( R

X

f dµ)g for some fixed g ∈ D and any f ∈ L

¹

(µ). Obviously, E = (E

^m,m+1

)

m≥0

∈ S , where for every m ∈ N

0

, E

^m,m+1

= E. Then for every m ∈ N

0

n→∞

lim k|P

_ε^m,n

− E|k = 0.

Therefore P

_ε

is uniformly asymptotically stable. Clearly, d

_n.^P

(P, P

_ε

) =

∞

X

m=M +1

1 2

^m+1

 P

^m,m+1

− E   ≤ 1

2

^M

< ε, which completes the proof.

We will now discuss a weaker case of asymptotic stability of chains of Markov operators, i.e., almost uniform asymptotic stability. We begin with

Definition. A nonhomogeneous chain of Markov operators P is said to be almost uniformly asymptotically stable if for every m ∈ N

0

n→∞

lim sup

f,g∈D

kP

^m,n

f − P

^m,n

gk

₁

= 0.

The set of all almost uniformly asymptotically stable Markov operators is denoted by S

a.uas

.

Repeating arguments from [1] or following the proof of Theorem 4.6 in [12] (cf.

[8]), we obtain a useful characterization of almost uniformly asymptotically stable nonhomogeneous chains of Markov operators.

Theorem 3. Let P ∈ S . If there exists a sequence (λ

n

)

_n∈N0

, 0 ≤ λ

n

< 1, satisfying

∞

X

n=0

λ

_n

= ∞

and such that for every f , g ∈ D we have P

^n,n+1

f ∧ P

^n,n+1

g

1

≥ λ

_n

for all n ∈ N

0

,

then P is almost uniformly asymptotically stable (here ∧ stands for the ordinary minimum in L

¹

(µ)).

Almost uniform asymptotic stability means that the influence of the state of the

process at the time m on its future states decreases (uniformly) to zero with the

passage of time. Thus, in the case of nonhomogeneous chains of Markov operators

this property is essentially weaker than the uniform asymptotic stability which

additionally claims the existence of a ”stationary” (common limiting) density. In the class of homogeneous chains of Markov operators notions of uniform asymp- totic stability and almost uniform asymptotic stability coincide. Indeed, if there exists ε > 0 such that for some n

0

and every f , g ∈ D we have kP

ⁿ⁰

f ∧P

ⁿ⁰

gk

1

≥ ε, then repeating arguments from [1] we obtain that kP

ⁿ⁰

f −P

ⁿ⁰

gk

1

≤ (1−ε)kf −gk

₁

and we conclude that the mapping P is a strict contraction. Applying the Banach fixed point theorem there exists a unique P - invariant density f

∗

∈ D such that lim

n→∞

kP

ⁿ

f − f

∗

k

₁

= 0, where f ∈ D is arbitrary. It follows that

sup

f ∈D

kP

ⁿ

f − f

∗

k

₁

≤ (1 − ε)

jn n0

k

· kf − f

_∗

k ≤ 2 (1 − ε)

jn n0

k

→ 0

uniformly for f ∈ D. Hence in the class of homogeneous chains of Markov operators

S

a.uas

= S

uas

= {P ∈ S : ∃

n

∃

_ε>0

∀

_f,g∈D

kP

ⁿ

f ∧ P

ⁿ

gk

₁

≥ ε} , which implies that in the homogeneous case the set S

a.uas

is norm open.

The following theorem states that almost uniformly asymptotically stable nonhomogeneous chains of Markov operators are generic. Its proof may be par- tially derived from Theorem 3, but for the convenience of the reader we give it in full detail.

Theorem 4. S

a.uas

is a dense G

_δ

subset of S in both sup norm and P norm topologies (i.e., in d

n. sup

and d

_n.^P

respectively).

Proof. First we will show that S

a.uas

is a d

_{n. sup}

dense subset of S (the denseness in the metric d

_n.^P

follows from the fact that d

_{n. sup}

is stronger than d

_n.^P

). To this end, given an arbitrary P ∈ S and 0 < ε < 1, consider a convex combination

P

_ε^m,m+1

= (1 − ε) P

^m,m+1

+ εE,

where as before E = (E

^m,m+1

)

_m≥0

∈ S is such that E

^m,m+1

= E for every m ∈ N

0

and Ef = ( R

X

f dµ)g for some fixed g ∈ D and any f ∈ L

¹

(µ). Clearly, P

ε

∈ S and d

n. sup

(P, P

_ε

) < 2ε. To prove that P

_ε

is almost uniformly asymptotically stable notice that for any densities f and g ∈ D we have

P

_ε^n−1,n

f − P

_ε^n−1,n

g

1

= (1 − ε)

P

^n−1,n

f − P

^n−1,n

g

1

= (1 − ε)

P

^n−1,n

(f − g)

1

≤ (1 − ε) kf − gk

₁

and therefore

kP

_ε^m,n

f − P

_ε^m,n

gk

₁

=

P

^n−1,n

P

^m,n−1

f − P

^m,n−1

g

1

≤ (1 − ε)

P

^m,n−1

f − P

^m,n−1

g

1

. Iterating the last inequality for any f, g ∈ D we have

kP

_ε^m,n

f − P

_ε^m,n

gk

₁

≤ (1 − ε)

^n−m

kf − gk

₁

. Hence

kP

_ε^m,n

f − P

_ε^m,n

gk

₁

≤ 2(1 − ε)

^n−m

for any f, g ∈ D. Thus,

sup

f,g∈D

kP

_ε^m,n

f − P

_ε^m,n

gk

₁

≤ 2 (1 − ε)

^n−m

.

Therefore,

n→∞

lim sup

f,g∈D

kP

_ε^m,n

f − P

_ε^m,n

gk

₁

= 0

and the denseness of the set S

a.uas

in S is proved for both sup norm and P norm topologies.

To show G

_δ

-ness of S

a.uas

observe that P

^m,n+1

f − P

^m,n+1

g

₁

=

P

^n,n+1

(P

^m,n

f ) − P

^n,n+1

(P

^m,n

g)

₁

≤ kP

^m,n

f − P

^m,n

gk

₁

,

which means that the sequence kP

^m,n

f − P

^m,n

gk

₁

is nonincreasing. It follows that for the fixed m the sequence sup

_f,g∈D

kP

^m,n

f − P

^m,n

gk

₁

is nonincreasing as well. We obtain that

S

a.uas

= (

P ∈ S : ∀

m∈N0

lim

n→∞

sup

f,g∈D

kP

^m,n

f − P

^m,n

gk

₁

= 0 )

=

∞

\

m=0

∞

\

k=1

∞

[

n=m+1

(

P ∈ S : sup

f,g∈D

kP

^m,n

f − P

^m,n

gk

₁

< 1 k

)

.

Note that for fixed m < n the function S 3 P 7→ sup

f,g∈D

kP

^m,n

f − P

^m,n

gk

₁

is d

_n.^P

continuous. Hence S

a.uas

is a G

_δ

set for the metric d

_n.^P

(and it is G

_δ

set for the stronger metric d

n. sup

).

The following example shows that (in contrast to the homogeneous case) in the class of nonhomogeneous chains of Markov operators the set S

a.uas

is not open (even though it is a dense G

_δ

subset of S ).

Example 5. As before, since the measure µ is σ-finite, there exists a sequence {B

_k

}, B

_k

∈ A, such that B

_i

∩B

_j

= ∅ for i 6= j and X = S

∞

k=0

B

k

, 0 < µ (B

k

) < ∞ for all k ∈ N

0

. Define

S

^m,m+1

f =

_m+1¹

Z

X

f dµ



g

_m+1

+ 

1 −

_m+1¹

 X

^∞

j=m+1

a

_j−m−1

(f ) · g

_j+1

for any f ∈ L

¹

(µ) and any g

j

∈ D such that supp g

_j

⊆ B

_j

and where a

j

(f ) = R

Bj

f dµ. Applying Theorem 3 we obtain that S ∈ S

a.uas

as for every f , h ∈ D we have kS

^m,m+1

f ∧ S

^m,m+1

hk

1

≥

_m+1¹

and P

∞

m=0 1

m+1

= ∞. Fix ε > 0 and choose M such that

_{M +1}²

< ε. Consider T ∈ S defined for f ∈ L

¹

(µ) as follows:

T

^m,m+1

f =

( S

^m,m+1

f, if m < M,

P

∞

j=m+1

a

j−m−1

(f ) · g

j+1

, if m ≥ M.

Notice that for m ≥ M we have sup

_f,h∈D

kT

^m,n

f − T

^m,n

hk

₁

= 2 (e.g. take f , h ∈ D such that supp f ⊆ B

M +1

and supp h ⊆ B

M +2

). Hence T / ∈ S

a.uas

. We easily find that d

n.sup

(S, T) ≤

_m+1²

< ε. It follows that S

a.uas

is not norm open.

3. Strong operator topology asymptotic stability

This section is dedicated to the study of the asymptotic stability of nonhomo- geneous chains of Markov operators in strong operator topology on S . Similar to the previous section, we introduce two types of limit behaviour with the only difference that the mode of convergence is strong. We begin with

Definition. A nonhomogeneous chain of Markov operators P is called strong

asymptotically stable if there exists (a unique) f

∗

∈ D such that for every m ∈ N

0

and every f ∈ D

n→∞

lim kP

^m,n

f − f

∗

k

₁

= 0.

The set of all strong asymptotically stable Markov operators is denoted by S

sas

. Obviously S

uas

⊆ S

sas

.

Theorem 6. The set S

sas^c

of all Markov operators which are not strong asymp- totically stable is P sup topology dense subset of S (i.e., in d

so. sup

). Moreover, S

_sas^c

contains P P dense G

_δ

set (i.e., in d

_so.^P

).

Proof. Similar arguments to those which were used towards the proof of The- orem 1 imply the first part of the above theorem. Therefore we only prove the second statement.

Let {f

₀

, f

₁

, . . .} be a fixed countable and linearly dense subset of D. As before, we find a sequence {B

k

} such that B

_k

∈ A, B

_i

∩ B

_j

= ∅ for i 6= j and X = S

∞

k=0

B

_k

, 0 < µ(B

_k

) < ∞ for all k ∈ N

0

. To see that S

sas^c

contains the P P dense G

_δ

set observe that

S

_sas^c

⊇ (

P ∈S :∀

t∈N0

∀

_m∈N0

∀

_j∈N0

∀

_ı∈N

∀

_{N ∈N0}

∃

n>max{N,m}

t

X

k=0

Z

B_k

P

^m,n

f

j

dµ < 1 ı )

=

∞

\

t=0

∞

\

m=0

∞

\

j=0

∞

\

ı=1

∞

\

N =0

[

n>max{N,m}

(

P ∈ S :

t

X

k=0

Z

Bk

P

^m,n

f

j

dµ < 1 ı

) .

Clearly, the mapping

S 3 P 7→

t

X

k=0

Z

Bk

P

^m,n

f

j

dµ

is d

_so.^P

continuous, hence the sets {P ∈ S : P

^t_k=0

R

Bk

P

^m,n

f

_j

dµ <

¹_l

} are open in the topology induced by d

_so.^P

. It remains to show that the set

(

P ∈ S : ∀

t∈N0

∀

_m∈N0

∀

_j∈N0

∀

_ı∈N

∀

_{N ∈N0}

∃

n>max{N,m}

t

X

k=0

Z

Bk

P

^m,n

f

j

dµ < 1 ı

) (∗)

is P P dense. Now then, let P ∈ S and 0 < ε < 1 be taken arbitrarily. There exists m

0

∈ N

0

such that

∞

X

l=0

∞

X

m=m0+1

1 2

^m+l+1

=

∞

X

l=0

1 2

^l+1

∞

X

m=m0+1

1 2

^m

= 1

2

^m0

< ε

2 .

Let g

k

∈ D be such that supp g

_k

= {x ∈ X : g

k

(x) 6= 0} ⊆ B

k

for any k ∈ N

0

. Consider P

ε

∈ S defined as follows:

P

_ε^m,m+1

f =

( (1 − ε) P

^m,m+1

f + ε R

X

f dµ
g

_m0+1

, if 0 ≤ m ≤ m

₀

, R

X

f dµ
g

_m+1

, if m > m

₀

for any f ∈ L

¹

(µ). We notice that Z

t

S

k=0

B_k

P

_ε^m,n

f dµ = 0

if n > max{m

0

+ 1, t}, so clearly P

ε

= (P

ε^m,m+1

)

m≥0

is an element of the consid- ered subset (∗). Moreover, we have

d

_so.^P

(P

ε

, P)

=

∞

X

m,l=0

1 2

^m+l+1

P

_ε^m,m+1

f

_l

− P

^m,m+1

f

_l

1

=

∞

X

l=0 m0

X

m=0

1 2

^m+l+1

P

_ε^m,m+1

f

_l

− P

^m,m+1

f

_l

1

+

∞

X

l=0

∞

X

m=m0+1

1 2

^m+l+1

P

_ε^m,m+1

f

l

− P

^m,m+1

f

l

1

=

∞

X

l=0 m0

X

m=0

1 2

^m+l+1

(1 − ε)P

^m,m+1

f

_l

+ ε

Z

X

f

_l

dµ



g

m0+1

− P

^m,m+1

f

_l

1

+

∞

X

l=0

∞

X

m=m0+1

1 2

^m+l+1

Z

X

f

l

dµ



g

m+1

− P

^m,m+1

f

l

1

= ε

∞

X

l=0 m0

X

m=0

1 2

^m+l+1

g

m0+1

− P

^m,m+1

f

_l

₁

+

∞

X

l=0

∞

X

m=m0+1

1 2

^m+l+1

g

m+1

− P

^m,m+1

f

_l

1

≤ ε

∞

X

l=0 m0

X

m=0

1

2

^m+l+1

· 2 +

∞

X

l=0

∞

X

m=m0+1

1 2

^m+l+1

· 2

< ε

∞

X

l=0 m0

X

m=0

1 2

^m+l

+ ε

2 · 2 = ε

 1 − 1

2

^m0



^∞

X

l=0

1

2

^l

+ ε < 2ε + ε = 3ε,

which completes the proof.

On the other hand, the fact that S

uas

⊆ S

sas

and the Proposition 2 lead to the following

Proposition 7. The set S

sas

is P P topology dense in S (i.e., in d

so.P

).

Let us proceed with

Definition. A nonhomogeneous chain of Markov operators P is called strong almost asymptotically stable if for every m ∈ N

0

and f , g ∈ D

n→∞

lim kP

^m,n

f − P

^m,n

gk

₁

= 0.

The set of all strong almost asymptotically stable Markov operators is denoted by S

a.sas

.

Clearly, S

a.uas

⊂ S

a.sas

. It should be emphasized that unlike the uniform case, in the class of homogeneous chains of Markov operators notions of strong asymptotic stability and strong almost asymptotic stability are essentially different (cf. [4]).

We easily obtain that strong almost asymptotically stable nonhomogeneous chains of Markov operators are generic.

Theorem 8. The set S

a.sas

is a dense G

_δ

subset of S in both P sup and P P strong operator topologies (i.e., in d

so. sup

and d

_so.^P

respectively).

Proof. It remains to show the G

_δ

-ness of S

a.sas

. Let {f

0

, f

1

, . . .} be a fixed countable and linearly dense subset of D. Notice that

S

s.aas

= n

P ∈ S : ∀

m∈N0

∀

_i∈N0

∀

_j∈N0

lim

n→∞

kP

^m,n

f

i

− P

^m,n

f

j

k

₁

= 0 o

=

∞

\

m=0

∞

\

i=0

∞

\

j=0

∞

\

ı=1

∞

\

N =0

[

n>max{N,m}



P ∈ S : kP

^m,n

f

_i

− P

^m,n

f

_j

k

₁

< 1 ı

 .

Observe that the sequence n 7→ kP

^m,n

f

_i

− P

^m,n

f

_j

k

₁

is nonincreasing and that for fixed m < n the function S 3 P 7→ kP

^m,n

f

i

− P

^m,n

f

j

k

₁

is continuous for the metric d

_so.^P

. Hence S

a.sas

is a G

_δ

set for the metric d

_so.^P

(and it is G

_δ

set for the stronger metric d

_{so. sup}

as well).

References

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1

,

Proc. Amer. Math. Soc. 133 (2005) 2119–2129.

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Roumaine Math. Pures Appl. 43 (1998) 375–392.

[6] M. Iosifescu, On two recent papers on ergodicity in nonhomogeneous Markov chains, Annals Math. Stat. 43 (1972) 1732–1736.

[7] M. Iosifescu, Finite Markov Processes and Their Applications (John Wiley and Sons, 1980).

[8] D.L. Isaacson and R.W. Madsen, Markov Chains: Theory and Applications (Wiley, New York, 1976).

[9] A. Iwanik and R. R¸ ebowski, Structure of mixing and category of complete mixing for stochastic operators, Ann. Polon. Math. 56 (1992) 233–242.

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₁

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Received 11 April 2012

Revised 23 February 2013